The Vector Maximization Problem: Proper Efficiency and Stability

Authors:
  • HBH. P. Benson
    H. P. Benson
  • TMT. L. Morin
    T. L. Morin
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Abstract
Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. While the concept of efficiency has played a useful role in the analysis of such problems, a slightly more restricted concept of efficiency, that of proper efficiency, has been proposed in order to eliminate efficient solutions of a certain anomalous type. In this paper, necessary and sufficient conditions for an efficient solution to be properly efficient are developed. These conditions relate the proper efficiency of a given solution to the stability of certain single-objective maximization problems. The conditions are useful both in verifying that certain efficient solutions are properly efficient and in identifying efficient solutions that are not proper. An immediate corollary of the theory is that all efficient solutions in linear vector maximization problems are properly efficient. Examples are given to illustrate our results.